The problem is worst in urban areas in England
The government is expected to announce an extra £200m this week to provide new primary school places in parts of England struggling with shortages.
London and the West Midlands have been hardest hit. Rising birth rates and the recession reducing numbers in private education are being blamed.
The shortage most affects places for four and five-year-olds.
A report earlier this year by London Councils said there was a shortfall of 2,250 places in this financial year.
That would rise to 5,000 next year and to 18,300 by 2014, the councils said.
The report said 25 of the capital's 33 authorities were experiencing capacity problems or were expected to within two to three years.
In England's worst-hit areas, councils have been building temporary classrooms or making plans to expand schools.
In Birmingham, there are plans for more than 1,400 new primary places.
The National Union of Teachers says too many schools were closed when the birth rate was falling, meaning there is no capacity in the system now.
The government has previously criticised councils, saying they had not planned ahead properly when population projections were available.
The London borough of Richmond says there was no way it could have predicted the 8% reduction in the number of children being educated privately.
A spokesman for the Department for Children Schools and Families said: "Local authorities are under a legal duty to ensure sufficient school places are available to meet parental demand across their areas - with the record capital investment in schools since 1997 based on their pupil projections.
"But we know that specific areas, including some in London, now say that they are dealing with unanticipated rises in demand for reception classes over the next few years.
"It is clear that some simply did not plan for the rise in local birth rates, despite having four years to provide the extra places needed.
"But we accept others face exceptional rises in demand for new primary places through a range of local circumstances, which will put great pressures on their capital funding over the next two years.
"We have been working closely with individual authorities and representative bodies including London Councils, the Association of London Directors of Children's Services, on how to assist those with the greatest needs - and we plan to set out the next steps in full to Parliament very shortly."